Abstract

The locally constrained inverse curvature flow in space forms was introduced by Brendle, Guan and Li and is a powerful tool to prove the Alexandrov-Fenchel inequalities. In the first part of the talk, we show that h-convexity is preserved by Brendle-Guan-Li’s flow in hyperbolic space and as applications we prove some optimal geometric inequalities for h-convex hypersurfaces.  In the second part, we will discuss the locally constrained inverse curvature flow for capillary hypersurfaces in the Euclidean half-space which was introduced by Wang,Weng and Xia. We develop the tool of tensor maximum principle with boundary and then apply it to show that convexity is preserved by Wang-Weng-Xia’s flow. As application, we obtain a complete family of Alexandrov-Fenchel type inequalities in half-space. The talk is based on joint work with Haizhong Li, Yingxiang Hu, Bo Yang, and Tailong Zhou.